On a 16-1700s ship, did the navigator have his own room/section of the ship to work? If so, where? Any articles I find just talk about how the navigate, not where.
Generally, for your time period and, indeed, into the 19th century, navigation on a naval vessel was the province of the ship's master, though the ship's captain would, or should, have the same skills. On smaller vessels the captain might serve both roles. For the basics see https://en.wikipedia.org/wiki/Master_(naval)
The master had his own berth, though like everyone else below the captain space was limited. In most ships, to spread a chart (ships use charts, not maps) a convenient table in the captain's cabin would usually suffice if contriving a great distance. Smaller scale could fit a fold down table in the master's cabin. most very early charts were kept in books called rutters. See https://en.wikipedia.org/wiki/Rutter_(nautical)
One usually does not spread out some great big chart, but rather has a library of coastal charts as it is more important not to bump into anything than plotting your way across, say, the Atlantic. Using the equipment of the day, one tries for a noon sighting for determining one's location, then, according to the destination, adjusting course accordingly. This helps accounting for leeway, the tendency for the wind to not only drive the ship, but also push it sideways as it does so, so several days of no noon sighting might require a larger course correction. One does not really need to drag out charts until one is with a day's sail of an expected landfall. Consistent noon sightings can be recorded until one reaches near a pre-determined point… that's when the charts come out.
As an officer of wardroom rank, the master would have access to the wardroom facilities on an equal basis with the other wardroom officers: the vessel's lieutenants, purser, and surgeon.
Also, notwithstanding Wikipedia's claim, navigation despite it's importance would have taken up only a small fraction of a master's time onboard. Square-rig sailing was on the open sea was about staying in favourable winds and currents, away from dangerous shores. Actual care of the vessel's rigging, and supervision of the rigging crew, would have occupied most of a master's day. Working jointly with the helmsman, keeping a sailing vessel well-trimmed is a constant task of riding the wind and waves to beneficial effect. Two articles by Sam Willis (respectively on windward performance and maneuverability of sailing warships) are the best informed description of the process that I am aware of.
My sailing experience is almost all on two-person sloop-rigged (single mast, fore-and-aft sails) dinghy; but the basic pattern is of the nature of sailing more than of the specific vessel. Off the wind a compass course can be steered by the helmsman and crew is responsible for trimming sails constantly to keep them filled. The master and his mate would have supervised that trimming constantly to keep the sails full. Only upwind, at which square riggers were almost incapable, would sails be set with the helmsman responsible for keeping sails full.